ℝikhav Shah
I'm a sixth year graduate student of Nikhil Srivastava. Starting Fall 2025, I will be a CLE Moore Instructor at MIT. My interests include theoertical computer science, numerical linear algebra, random matrix theory, and probability.
Papers
The Pseudospectrum of Random Compressions of Matrices
Bounds are given for the area of the pseudospectrum of a randomly chosen compression of a given matrix. (external links: arXiv)
Sparse Pseudospectral Shattering
Regularization of the eigenvector condition number by perturbing just O(n log^2(n)) entries. (external links: arXiv)
A Kaczmarz-Inspired Method for Orthogonalization
Given a set of n vectors in n dimensions, how many times do you have to orthogonalize two randomly selected vectors against each other before the entire set becomes orthonormal? (external links: arXiv)
Fast Hermitian Diagonalization with Nearly Optimal Precision
An algorithm for diagonalizing Hermitian matrices in matrix multiplication time using just log(1/epsilon) + O(log(n)) bits. (external links: arXiv, SODA 2025)
Achieving the Highest Possible Elo Rating
If n chess players starting with equal rating play a total of k games, how high could one possibly be rated after the games? (external links: arXiv, FUN 2024)
A Spectral Approach to Polytope Diameter
Upper bounds on graph diameter of a polytope in two settings: (1) worst-case for integer polytopes with short bit description (2) smoothed unit polytopes. slides. (external links: arXiv, ITCS 2022, talk)
Determinants of binary matrices achieve every integral value up to Omega(2^n/n)
Exactly as the title states. An explicit construction is provided. (external links: arXiv, Linear Algebra and its Applications 2022)
Smoothed analysis of the condition number under low-rank perturbations
Adding a random rank k matrix to a fixed matrix of rank n-k produces a well-conditioned matrix. (external links: arXiv, RANDOM 2021)
Directed Random Geometric Graphs
Novel class of random digraphs for modeling real-world directed networks. (external links: arXiv, Journal of Complex Networks)
Posters and Talks
Joint Mathematics Meeting 2025
Regularization of the eigenvector condition number via sparse perturbations. What is the effect of perturbing a small number of entries of a matrix on the conditioning of its eigenvectors and stability of its eigenvalues?
Joint Mathematics Meeting 2024
Subspace iteration for nonnormal eigenvalue problems. A variant of subspace iteration has a provable runtime depending on the inverse square of a non-consecutive eigenvalue gap.
Innovations in Theoretical Computer Science 2022
Presentation of ''A Spectral Approach to Polytope Diameter''
Opt-ML (NeurIPS satellite)
Using random projections to speed up t-SNE without degrading clustering accuracy. poster.
Sloan Sports Analytics Conference
Treated basketball players as points traveling relative to a gradient of the `basket probabiltiy' function which assigns the probability of making a shot to each location on the court.
Applied Projects
Statistical Paradoxes
Interactive demonstrations of why seemingly obvious conclusions don't necessarily follow when examining real-world data.
Probabilistic Counters
Analyzing performance of probabilisitc counters in theory and practice; addressing an apparent bug in the Redis caching system. (external links: GitHub issue)
Producing Formula Representations of Mathematical Text (Patented)
Algorithm for parsing English transcriptions of mathematical formulas and converting that into, e.g., executable python code or LaTeX.
Hackathons
HackMIT 2018
Created graphical user interface for constructing & training neural networks, with fast iteration of network architecture and ability to export in industry standard format. [First Place Grand Prize Winner]
HackMIT 2017
Created image enhancer that increases resolution and recovers detail from pixelated images without using machine learning or image databases. [First Place Grand Prize Winner]
NBA Hackathon
Created tool to examine player matchups and optimal defender location on court throughout the game. [Third place winner]
Fun
Intransitive game
There's an elegant characterization of every symmetric, zero-sum game, which may be intransitive in general (e.g. rock-paper-scissors). In particular, all payoffs are determined by a difference between two simple inner products. I've implemented a visualization of a random instance of such a game being played between many players; I also wrote a brief proof that the characterization always works.
Boids
Classic boids simulation written in js and rendered using svg. Interestingly, not a single 'boid' implementation I was able to find online used the two key details outlined in the original SIGGRAPH paper. Namely, strength of repulsion is given by an inverse square law, and accleration is determined based on prioritized allocation. I find they dramatically improve the simulation!
Snake
Bot plays snake on different sized grids. The bot maintains a cycle cover of the grid graph such that the snake is a subset of the edges in the cover. The cycle cover is modified each step to attempt to put the snake in the same cycle as the food, and to make that said cycle as short as possible.
Puzzles
Some puzzle-hunt style puzzles I made.
Bird game
A text-based game in which you're a bird and have to figure out your goals.
Also check out these people! Sachin Shah, Sandeep Silwal, Vilas Winstein.